My understanding of the purpose of the trope (cantillation) notes is that they were initially (Massorah) meant for the purposes of grammar and pronunciation. I've hunted a few web sites, and I couldn't locate anything that states that by Massorah (I.e. when given at Mt. Sinai) these notes had music to them.
I did locate in Talmud Megillah 32a:
ואמר ר' שפטיה אמר ר' יוחנן כל הקורא בלא נעימה ושונה בלא זמרה עליו הכתוב אומר (יחזקאל כ, כה) וגם אני נתתי להם חוקים לא טובים וגו'
R. Sheftiah further said in the name of R. Yohanan: If one reads [Scripture] without a melody or recites the Mishnah without a tune, of him the verse says, “For I gave them also statutes that were not good.” (Ezekiel 20:25).
(Note: I am aware that immediately afterwards, Abaye disputes this statement.)
By making this statement, R. Sheftiah seems to imply that one should sing the Torah reading (See Rashi's explanation, there) stating that he means using ta'amei hamikrah - cantillation notes).
What's unclear from the Gemara's statement are these areas:
- Is a person obligated ever to sing the notes whether he is reading Torah aloud individually or a congregational setting?
- If this is, in fact an obligation, did this obligation of music (any music whatsoever; I know that each community developed their own music for the notes) originate from Sinai or is it a Rabbinical obligation that came years later?